1. FIN303 Vicentiu Covrig 1 Time value of money (chapter 5)

2. FIN303 Vicentiu Covrig 2 Time line Show the timing of cash flows. Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period. CF 0 CF 1 CF 3 CF 2 0123 i%

3. FIN303 Vicentiu Covrig 3 Drawing time lines: $100 lump sum due in 2 years; 3-year $100 ordinary annuity 100 0123 i% 3 year $100 ordinary annuity 100 012 i% $100 lump sum due in 2 years

4. FIN303 Vicentiu Covrig 4 Drawing time lines: Uneven cash flow stream; CF 0 = -$50, CF 1 = $100, CF 2 = $75, and CF 3 = $50 100 50 75 0123 i% -50 Uneven cash flow stream

5. FIN303 Vicentiu Covrig 5 What is the future value (FV) of an initial $100 after 3 years, if I/YR = 10%? Finding the FV of a cash flow or series of cash flows when compound interest is applied is called compounding. FV can be solved by using the arithmetic or financial calculator FV = ? 0123 10% 100

6. FIN303 Vicentiu Covrig 6 After 1 year: - FV 1 = PV ( 1 + i ) = $100 (1.10) = $110.00 After 3 years: - FV 3 = PV ( 1 + i ) 3 = $100 (1.10) 3 =$133.10 After n years (general case): FV n = PV ( 1 + i ) n Using financial calculator: Requires 4 inputs, and will solve for the fifth. (Set to P/YR = 1 and END mode.) INPUTS OUTPUT NI/YRPMTPVFV 3100 133.10 -100

7. FIN303 Vicentiu Covrig 7 Present Value What is the present value (PV) of $100 due in 3 years, if I = 10%? I=10%; FV=$100; N=3; PV=? Finding the PV of a cash flow or series of cash flows when compound interest is applied is called discounting (the reverse of compounding). The PV shows the value of cash flows in terms of today’s purchasing power. PV = FV n / ( 1 + i ) n PV = FV 3 / ( 1 + i ) 3 = $100 / ( 1.10 ) 3 = $75.13

8. FIN303 Vicentiu Covrig 8 The calculator method: Exactly like solving for FV, except we have different input information and are solving for a different variable. INPUTS OUTPUT NI/YRPMTPVFV 3100100 -75.13

9. FIN303 Vicentiu Covrig 9 Solving for N: If sales grow at 20% per year, how long before sales double? Same as previous problems, but now solving for N. INPUTS OUTPUT NI/YRPMTPVFV 3.8 2002

10. FIN303 Vicentiu Covrig 10 Solving for I: what interest rate would cause $100 to grow to $125.97 in 3 years? Solves the general FV equation for I. INPUTS OUTPUT NI/YRPMTPVFV 3 8 0 125.97 -100

11. FIN303 Vicentiu Covrig 11 What is the difference between an ordinary annuity and an annuity due? Ordinary Annuity (payments at the end of the period) PMT 0123 i% PMT 0123 i% PMT Annuity Due (payments at the beginning of the period)

12. FIN303 Vicentiu Covrig 12 Solving for FV: 3-year ordinary annuity of $100 at 10% $100 payments occur at the end of each period, but there is no PV. INPUTS OUTPUT NI/YRPMTPVFV 310-100 331 0

13. FIN303 Vicentiu Covrig 13 Solving for PV: 3-year ordinary annuity of $100 at 10% $100 payments still occur at the end of each period, but now there is no FV. INPUTS OUTPUT NI/YRPMTPVFV 3101000 -248.69

14. FIN303 Vicentiu Covrig 14 What is the PV of this uneven cash flow stream? 0 100 1 300 2 3 10% -50 4 100/(1+0.1) = 90.91 300/(1+0.1)^2 = 247.93 300/(1+0.1)^3 = =225.39 -50/(1+0.1)^4 = -34.15 530.08 = PV

15. FIN303 Vicentiu Covrig 15 The Power of Compound Interest A 20-year-old student wants to start saving for retirement. She plans to save $1,095 a year (end-of-the-year investment deposit) in an online stock account. The stock account has an expected annual return of 12%. How much money will she have when she is 65 years old?

16. FIN303 Vicentiu Covrig 16 Solving for FV: Savings problem If she begins saving today, and sticks to her plan, she will have $1,487,261.89 when she is 65. INPUTS OUTPUT NI/YRPMTPVFV 4512-1095 1,487,262 0

17. FIN303 Vicentiu Covrig 17 Solving for FV: Savings problem, if you wait until you are 40 years old to start If a 40-year-old investor begins saving today, and sticks to the plan, he or she will have $146,000.59 at age 65. This is $1.3 million less than if starting at age 20. Lesson: It pays to start saving early. INPUTS OUTPUT NI/YRPMTPVFV 2512-1095 146,001 0

18. FIN303 Vicentiu Covrig 18 What is the PV of this uneven cash flow stream? 0 100 1 300 2 3 10% -50 4 90.91 247.93 225.39 -34.15 530.08 = PV

19. FIN303 Vicentiu Covrig 19 Will the FV of a lump sum be larger or smaller if compounded more often, holding the stated I% constant? LARGER, as the more frequently compounding occurs, interest is earned on interest more often. Annually: FV 3 = $100(1.10) 3 = $133.10 0 123 10% 100133.10 Semiannually: FV 6 = $100(1.05) 6 = $134.01 0123 5% 456 134.01 123 0 100

20. FIN303 Vicentiu Covrig 20 Classifications of interest rates Nominal rate (i NOM ) – also called the quoted or state rate or APR. An annual rate that ignores compounding effects. - i NOM is stated in contracts. Effective (or equivalent) annual rate (EAR = EFF%) – the annual rate of interest actually being earned, taking into account compounding. - EAR% for 10% semiannual investment EAR%= ( 1 + i NOM / m ) m - 1 = ( 1 + 0.10 / 2 ) 2 – 1 = 10.25% - An investor would be indifferent between an investment offering a 10.25% annual return and one offering a 10% annual return, compounded semiannually.

21. FIN303 Vicentiu Covrig 21 Why is it important to consider effective rates of return? An investment with monthly payments is different from one with quarterly payments. Must put each return on an EAR% basis to compare rates of return. Must use EAR% for comparisons. See following values of EAR% rates at various compounding levels. EAR ANNUAL 10.00% EAR QUARTERLY 10.38% EAR MONTHLY 10.47% EAR DAILY (365) 10.52%

22. FIN303 Vicentiu Covrig 22 When is each rate used? I NOM : Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines. I PER : Used in calculations and shown on time lines. If M = 1, I NOM = I PER = EAR. EAR: Used to compare returns on investments with different payments per year. Used in calculations when annuity payments don’t match compounding periods.

23. FIN303 Vicentiu Covrig 23 What is the FV of $100 after 3 years under 10% semiannual compounding? Quarterly compounding?

24. FIN303 Vicentiu Covrig 24 Can the effective rate ever be equal to the nominal rate? Yes, but only if annual compounding is used, i.e., if M = 1. If M > 1, EFF% will always be greater than the nominal rate.

25. FIN303 Vicentiu Covrig 25 What’s the FV of a 3-year $100 annuity, if the quoted interest rate is 10%, compounded semiannually? Payments occur annually, but compounding occurs every 6 months. Cannot use normal annuity valuation techniques. 01 100 23 5% 45 100 6

26. FIN303 Vicentiu Covrig 26 Method 1: Compound Each Cash Flow FV 3 = $100(1.05) 4 + $100(1.05) 2 + $100 FV 3 = $331.80 110.25 121.55 331.80 01 100 23 5% 45 100 6

27. FIN303 Vicentiu Covrig 27 Method 2: Financial Calculator or Excel Find the EAR and treat as an annuity. EAR = (1 + 0.10/2) 2 – 1 = 10.25%. 5- 27 INPUTS OUTPUT NI/YRPMTPVFV 3-100 10.25 0 331.80

28. FIN303 Vicentiu Covrig 28 Loan amortization Amortization tables are widely used for home mortgages, auto loans, business loans, retirement plans, etc. EXAMPLE: Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments. Step 1: find the required annual payment All input information is already given, just remember that the FV = 0 because the reason for amortizing the loan and making payments is to retire the loan. INPUTS OUTPUTNI/YR PMT PV FV 310 402.11 0-1000

29. FIN303 Vicentiu Covrig 29 Step 2:Find the interest paid in Year 1 The borrower will owe interest upon the initial balance at the end of the first year. Interest to be paid in the first year can be found by multiplying the beginning balance by the interest rate. INT t = Beg bal t (i) INT 1 = $1,000 (0.10) = $100 Step 3:Find the principal repaid in Year 1 If a payment of $402.11 was made at the end of the first year and $100 was paid toward interest, the remaining value must represent the amount of principal repaid. PRIN= PMT – INT = $402.11 - $100 = $302.11

30. FIN303 Vicentiu Covrig 30 Step 4:Find the ending balance after Year 1 To find the balance at the end of the period, subtract the amount paid toward principal from the beginning balance. END BAL= BEG BAL – PRIN = $1,000 - $302.11 = $697.89 Constructing an amortization table: Repeat steps 1 – 4 until end of loan Interest paid declines with each payment as the balance declines. What are the tax implications of this?

31. FIN303 Vicentiu Covrig 31 Loan amortization/ Mortgage EXAMPLE: You have just bought a new house and have a 30-year fixed rate mortgage of $300,000, at an interest rate of 6% (compounded monthly). What are your monthly payments? How much of your first month’s payment is principal and how much is interest? A. Monthly payment: $1,799 B. Interest= $1,500

32. FIN303 Vicentiu Covrig 32 Learning objectives All the applications and concepts covered in the PowerPoint notes The numerical problems on the exam will be similar to the ones in the notes How does the present value of an amount in the future change as the time is extended and as the interest rate increases? DO NOT need to prepare Annuity Due, section 5-17 for the exam. Need to learn how to use the financial calculator on your own. Recommended end-of-chapter problems: ST-1 to ST-4, 5-1 to 5-19, 5-22 to 5-24